%I #16 Apr 08 2020 07:31:23
%S 0,0,0,0,105,2625,42075,554820,6578550,73169250,781319370,8122058850,
%T 82922497890,836339477160,8366154130425,83235403604220,
%U 825247074255735,8165187992777430,80704316324506860,797435573602269015,7881226621071221625,77939438593071188565
%N Number of rooted binary leaf-multilabeled trees with n leaves on the label set [5].
%H Andrew Howroyd, <a href="/A220822/b220822.txt">Table of n, a(n) for n = 1..200</a>
%H V. P. Johnson, <a href="http://people.math.sc.edu/czabarka/Theses/JohnsonThesis.pdf">Enumeration Results on Leaf Labeled Trees</a>, Ph. D. Dissertation, Univ. Southern Calif., 2012.
%p b:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,
%p (t-> t*(1-t)/2)(b(n/2, k)))+add(b(i, k)*b(n-i, k), i=1..n/2))
%p end:
%p a:= n-> (k-> add((-1)^i*binomial(k, i)*b(n, k-i), i=0..k))(5):
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Sep 07 2019
%t b[n_, k_] := b[n, k] = If[n < 2, k n, If[OddQ[n], 0, Function[t, t (1 - t)/2][b[n/2, k]]] + Sum[b[i, k] b[n - i, k], {i, 1, n/2}]];
%t a[n_] := Function[k, Sum[(-1)^i Binomial[k, i] b[n, k - i], {i, 0, k}]][5];
%t Array[a, 30] (* _Jean-François Alcover_, Apr 08 2020, after _Alois P. Heinz_ *)
%Y Column 5 of A319541.
%K nonn
%O 1,5
%A _N. J. A. Sloane_, Dec 22 2012
%E Terms a(11) and beyond from _Andrew Howroyd_, Sep 23 2018